57.5.18 problem 3(f)

Internal problem ID [14373]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 3(f)
Date solved : Thursday, October 02, 2025 at 09:34:15 AM
CAS classification : [_linear]

\begin{align*} R^{\prime }&=\frac {R}{t}+t \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} R \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 16

ode:=diff(R(t),t) = R(t)/t+t*exp(-t); 
ic:=[R(1) = 1]; 
dsolve([ode,op(ic)],R(t), singsol=all);
 

\[ R = \left (-{\mathrm e}^{-t}+1+{\mathrm e}^{-1}\right ) t \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 19

ode=D[ R[t],t]==R[t]/t+t*Exp[-t]; 
ic={R[1]==1}; 
DSolve[{ode,ic},R[t],t,IncludeSingularSolutions->True]
 

\begin{align*} R(t)&\to \left (-e^{-t}+1+\frac {1}{e}\right ) t \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
R = Function("R") 
ode = Eq(-t*exp(-t) + Derivative(R(t), t) - R(t)/t,0) 
ics = {R(1): 1} 
dsolve(ode,func=R(t),ics=ics)
 

\[ R{\left (t \right )} = t \left (\frac {1 + e}{e} - e^{- t}\right ) \]